A generalisation of the Euclid-Mullin sequences

TL;DR

This paper generalizes Euclid-Mullin prime sequences to include primes in specific residue classes using cyclotomic polynomials, proving infinite prime omissions under certain hypotheses and unconditionally for some cases.

Contribution

It extends Mullin's prime-generating sequences to residue classes and demonstrates infinite prime omissions under the Extended Riemann Hypothesis and unconditionally for some moduli.

Findings

Sequences omit infinitely many primes in certain residue classes under RH.

Unconditionally, at least one prime is omitted for infinitely many moduli.

Generalizes previous results for specific moduli to a broader class.

Abstract

We extend Mullin's prime-generating procedures to produce sequences of primes lying in given residue classes. In particular we study the sequences generated by cyclotomic polynomials $\Phi_m(cx)$ for suitable $c\in\mathbb{Z}$. Under the Extended Riemann Hypothesis in general and unconditionally for some moduli, we show that the analogue of the second Euclid--Mullin sequence omits infinitely many primes $\equiv1\pmod{m}$. We further show unconditionally that at least one prime is omitted for infinitely many $m$. This generalises work of the first author for $m=1$ and the second author for $m=2^k$.